A Common View on Strong, Uniform, and Other Notions of Equivalence in Answer-Set Programming

Logic programming under the answer-set semantics nowadays deals with numerous different notions of program equivalence. This is due to the fact that equivalence for substitution (known as strong equivalence) and ordinary equivalence are different concepts. The former holds, given programs P and Q, iff P can be faithfully replaced by Q within any context R, while the latter holds iff P and Q provide the same output, that is, they have the same answer sets. Notions in between strong and ordinary equivalence have been introduced as theoretical tools to compare incomplete programs and are defined by either restricting the syntactic structure of the considered context programs R or by bounding the set A of atoms allowed to occur in R (relativized equivalence).For the latter approach, different A yield properly different equivalence notions, in general. For the former approach, however, it turned out that any ``reasonable'' syntactic restriction to R coincides with either ordinary, strong, or uniform equivalence. In this paper, we propose a parameterization for equivalence notions which takes care of both such kinds of restrictions simultaneously by bounding, on the one hand, the atoms which are allowed to occur in the rule heads of the context and, on the other hand, the atoms which are allowed to occur in the rule bodies of the context. We introduce a general semantical characterization which includes known ones as SE-models (for strong equivalence) or UE-models (for uniform equivalence) as special cases. Moreover,we provide complexity bounds for the problem in question and sketch a possible implementation method. To appear in Theory and Practice of Logic Programming (TPLP)

On graph equivalences preserved under extensions

Let R be an equivalence relation on graphs. By the strengthening of R we mean the relation R' such that graphs G and H are in the relation R' if for every graph F, the union of the graphs G and F is in the relation R with the union of the graphs H and F. We study strengthenings of equivalence relations on graphs. The most important case that we consider concerns equivalence relations defined by graph properties. We obtain results on the strengthening of equivalence relations determined by the properties such as being a k-connected graph, k-colorable, hamiltonian and planar

Strong Equivalence and Program's Structure in Arguing Essential Equivalence between Logic Programs

Answer set programming is a prominent declarative programming paradigm used in formulating combinatorial search problems and implementing distinct knowledge representation formalisms. It is common that several related and yet substantially different answer set programs exist for a given problem. Sometimes these encodings may display significantly different performance. Uncovering {\em precise formal} links between these programs is often important and yet far from trivial. This paper claims the correctness of a number of interesting program rewritings